joint binary response modeling for childhood comorbidity
TRANSCRIPT
Joint Binary Response Modeling for Childhood Comorbidity in Ethiopia
Tesfaye Abera Bokoro1* Habtamu Kiros Gebresilasie1†
1Department of Statistics, Haramaya University, Dire Dawa, Ethiopia
Abstract
Background: Childhood diarrhea and Acute Respiratory Infection (ARI) are some of the
diseases that share common risk factors in tropical developing regions. The objective of the
study was to identify risk factors of childhood diarrhea and ARI among children age under-
five years old based on 2016 Ethiopian Demographic and Health Survey data.
Methods: A joint binary response model that accommodates the interdependence between
the two diseases was employed.
Results: We found a common odds ratio value (4.30) greater than unity, describing a positive
association between the two childhood diseases. Thereby, employing a joint model to assess
the potential factors for diarrhea and acute respiratory infection (ARI) was reasonable.
Moreover, it was identified that standard errors of the parameter estimates in the joint
response model were smaller compared to the corresponding standard errors of the separate
models.
Conclusion: In the joint model, explanatory variables such as residence, vaccination,
mother’s education, and antenatal care visits during pregnancy were found statistically
significant risk factors for diarrhea, whereas residence, number of children ever born,
vaccination, mother’s education, and wealth index were statistically significant risk factors
for childhood Acute Respiratory Infection. The two correlated dichotomous response
variables that is, diarrhea and ARI were affected together significantly by the risk factors
such as residence, vaccination, and mother’s education.
Key words: Diarrhea; ARI; bivariate binary response; comorbidity; joint modelling
1. Introduction
Morbidity in children living in low-income countries is commonly characterized by more
than one health condition [11], and is a challenge in many settings that may lead to death. On
the other hand, the problem of coinfection due to overlapping risk factors such as nutrition,
sanitation, and overcrowding [9] is highly prevalent among children under five years seeking
care [10]. Specifically, diarrhea and ARI (Acute Respiratory Infection) are some of the
diseases that share common risk factors in tropical developing regions [22].
Diarrhea is one of the key contributors to deaths for under–five children in Ethiopia.
According to, the World Health Organization (WHO) estimates, diarrhea contributes to more
than 13% (one in every ten) child deaths in Ethiopia [5]. The trend from previous studies
depicts, percentage of children under age 5 who had diarrhea decreased from 24% in 2000 to
12% in 2016 [3]. Furthermore, acute respiratory infection (ARI), and especially pneumonia,
is one of the leading causes of morbidity and mortality that accounts for 18% of deaths in
Ethiopia [5]. The country has made remarkable commitment to reduce the childhood
morbidity and mortality of ARI [4]. Deaths can be reduced if health interventions are targeted
at either one or more simultaneous diseases [11]. Usually programs for child care addressed
single diseases such as diarrhea, acute respiratory, fever, and malaria infections [10].
Diarrhea and acute respiratory infections remain the leading cause of death and often co-
occur in children under the age of 5 years old [24]. The illness of the two health conditions
may be as a result of shared risk factors at childhood, or as a result of shared external factors.
For instance, diarrhea and acute respiratory diseases may both share risk factors such as age,
as a child-dependent risk factor, or poor sanitation and crowding as the environmental risk
factors [25].
For child care programs under health institutions to be successful, statistical tools are
essential to determine the characteristics of diseases that coexist. In recent time, a joint
analysis that examines risk factors of such diseases has become more popular in identifying
similar patterns of variation [26]. The joint modeling addresses the correlation of the two
responses variables with the same set of covariates. The two outcomes provide better control
over the type I error rates in multiple tests and increases efficiency in the parameter estimates.
Most of previously conducted health researches in Ethiopia have emphasized on studying
a single disease at a time and joint response modeling of correlated outcomes focused on
continuous, discrete or a mixture of discrete and continuous outcomes [1].
However, due to common and overlapping risk factors, separate analyses may fail to provide
a comprehensive picture of the epidemiology of the diseases and the combined effects of
childhood diseases on the population under consideration [15].
Epidemiological methodology and research have increased over the years, and have
extended from studying a single disease to several diseases jointly at a time. Thus, in this
paper, we specifically aimed at comorbidity among under-five children using joint response
model that accommodates the interdependence between the two infections in assessing their
risk factors, so that the government of Ethiopia and interested institutions engaged in the
health sectors are able to use the findings, for better design and implementation of
intervention strategies that address exposure to multiple illnesses in the country.
2. Methodology
2.1 Data source
The empirical analysis in this paper is based on data available from the 2016 Ethiopian
Demographic and Health Survey (EDHS) .The Demographic and Health Surveys (DHS) are a
well-established source of reliable population data with a substantial focus on childhood diseases
[3]. There were about 41,392 children under the age of five years records in the 2016 survey of
Ethiopia. Each record consists of information on childhood diseases and the list of covariates that
could affect the health status of children. However, to reduce possible bias in the conclusion,
totally 9917 children were considered by omitting those cases with the missing data in this study.
2.2 Description of Study variables
Let Y be a vector of two dichotomous dependent variables, that is Y1 (Diarrhea Status) and Y2
(Acute Respiratory Infection Status). If the child had diarrhea and acute respiratory infection,
the two binary response variables would assume each a value ‘1’ or ‘0’ otherwise. The possible
joint outcomes of Y1 and Y2 for a set of m-paired observations with their probabilities of
occurrence can be illustrated as in Table1 below.
Table1: Possible combination of outcomes with corresponding probabilities of occurrence
Acute Respiratory Infection (Y2)
Total Diarrhea
(Y1)
y1=0
y2=0 y2=1
p00 p01 1-p1
y1=1 p10 p11 p1
Total 1-p2 p2 1
Similarly, we can represent the joint and separate probabilities of Y1 and Y2 indicated in the
table1 as, pij=P( y1= i, y2= j); i,j = 0,1 and pk = P(yk=1) , k = 1,2 respectively.
Where:
P11 = the child has both diarrhea and ARI
P01 = the child has no diarrhea but has ARI
P10 = the child has diarrhea but not ARI
P00 = the child has neither diarrhea nor ARI
In the final models, the following predictors were included: child’s age in months, child’s sex,
residence, antenatal care visit, vaccination, mother’s education, number of children, and wealth
index.
2.3.Statistical models
2.3.1. Separate and joint models
Bivariate logistic regression model is not identical application wise to the ordinary binary logistic
regression model, since the primary objective in binary regression model is to predict the
categories of the response variable as function of covariates involved in the model. However, in
bivariate logistic regression model we try to examine the relationship between two different
dichotomous response variables (Y1 and Y2), by modelling jointly as a function of explanatory
variables under consideration. As depicted in table1, each pair of dependent
variables (Yi1,Yi2) has four possible outcomes, (Yi1=1,Yi2=1), (Yi1=1,Yi2=0), (Yi1=0,Yi2=1),
and (Yi1=0,Yi2=0).
The joint probability for each of these four outcomes is modeled with three systematic
components, these are:
The marginal (separate) distributions, that is P(Yi1=1) and P(Yi2=1)
The odds ratio, which is denoted by 𝜓 describes the dependence of one marginal
distribution on the other.
Bivariate logistic regression model or bivariate logistic odds-ratio model proposed by
[23, 27, 28, 32] is specified by modeling the marginal distribution of each of Yj , and the common
odds ratio. The odds ratio or 𝝍 is defined as the ratio of the odds of Y1=1 given that Y2=1 and
the odds of Y1=1 given that Y2= 0, that is 𝝍 =𝑝11𝑝00
𝑝10𝑝01 . The odds ratio is used to measure the
association between the two dichotomous response variables (Y1 and Y2); the value of 𝝍 equal
to one indicates lack of association between Y1 and Y2.
Application of bivariate logistic regression provides adjusted estimates of concordance
through simultaneous estimation of covariate effects on the odds ratio that describes the pair wise
association structure. Furthermore, adopting bivariate logistic regression analysis may offer
greater precision than unadjusted estimates obtain from other possible categorical distributions.
The influence of the independent variables on the marginal probabilities that is, marginal
probability of diarrhea (Y1) and ARI (Y2), and odds ratio can be expressed using regression
models. Likewise, the association between the covariates and each disease can be examined using
separate logistic regression models for each outcome:
logit[E(Y1j)] = logit[Pr(Y1j = 1|X1j, β1)] = β1TX1j (1)
logit[E(Y2j)] = logit[Pr(Y2j = 1|X2j, β2)] = β2TX2j (2)
These standard logistic regression models, indicated in equation (1) and 2, do not consider the
correlation between the two childhood diseases. However, we assume a set of unobserved
random effects are shared by the two diseases of the same individual, that is, the two diseases
share common unobservable features. Let 𝑢𝑗 denote the random intercept shared by the two
diseases of the jth (j=1,2,…, m) individual. Let 𝜙1𝑖 and 𝜙2𝑖 define dummy variables with 𝜙1𝑖 =
1 for i=1 and 𝜙2𝑖 = 1 for i=2. Then, the joint response model for the bivariate logit is given by
logit[E(Yij|uj) = logit[Pr (Yij = 1|Xij, β̃i, uj)]
= ϕ1j (β̃1T
X1j + uj) + ϕ2j(β̃2T
X2j + uj) (3)
Hence, in this equation, the bivariate responses (Y1j,Y2j) of all individuals are stacked into a
single response vector. The random intercepts are assumed to vary independently from one
individual to another. In addition, the random intercepts are assumed to be normally distributed
with zero mean and variance 𝜎𝑢2. Let G denote the distribution of 𝑢𝑗 . The joint response of an
individual is assumed to be independent given the shared random intercept. By using conditional
independence from this assumption, we can write the likelihood function of the joint response
model as follows:
𝐿(. ) = ∏ Pr (𝑌𝑖𝑗 = 1|𝑋𝑖𝑗, 𝛽𝑖, 𝑢𝑗)𝑚𝑗=1
=∏ {∫ Pr (𝑌𝑖𝑗 = 1|𝑋𝑖𝑗, 𝛽𝑖, 𝑢𝑗)𝜙𝐺(𝑢𝑗)} 𝑚𝑗=1
=∏ {∫ ∏ Pr (𝑌𝑖𝑗 = 1|𝑋𝑖𝑗, 𝛽𝑖, 𝑢𝑗)𝜙𝐺(𝑢𝑗)2𝑖=1 }𝑚
𝑗=1
=∏ {∫ ∏𝑒
𝑢𝑗+�̃�1𝑇
𝑋𝑖𝑗
1+𝑒𝑢𝑗+�̃�1
𝑇𝑋𝑖𝑗
𝜙𝐺(𝑢𝑗)2𝑖=1 }𝑚
𝑗=1 (4)
Following Muse (2015), we used maximum likelihood estimation to obtain estimates of the
model parameters (𝛽1,𝛽2, 𝜎𝑢2). The estimation was based on the maximization of the log-
likelihood function i.e. maximizing the logarithm of the likelihood function. However, since
the integrals do not have closed-form solutions for equation (4), parameters of the joint model
were estimated simultaneously via maximum likelihood by evaluating the integrals using
Gaussian adaptive quadrature approximation [17].
3. Results and Discussion
Analyses of this study were based on total valid observations of 9917 children under the age
of five years. As provided in Table 2, 5.9% and 8.2% of the male child and also 5.1% and
7.8% of the female child diarrhea and ARI cases were reported respectively in the past fifteen
days. Geographically, 4.6% of the reported childhood diarrhea and ARI lived in urban areas,
as compared to 22.1% who lived in rural areas. Regarding vaccination, 63.3% of the children
in Ethiopia were reported as received various types of vaccination. The covariates such as
residence, vaccination, antenatal visit, wealth index, mother’s education, birth order, drink
water source and region with (P<0.05) were found significantly associated with diarrhea and
ARI. Regionally, a low prevalence of childhood diarrhea was observed in the nation’s capital
Addis Ababa that could be due to the relative socioeconomic development advantages in
terms of access to health services, and a relatively higher prevalence of childhood diarrhea in
Oromia and SNNP regions but each disease has its typical geographical pattern of variation.
Table 3 also indicates that the prevalence of 4.3% childhood comorbidity of diarrhea and
ARI noted in Ethiopia in 2016 DHS. On the other hand, 11.7% of the children had only ARI
and 6.7% of the children had only diarrhea morbidity.
Table 2: Distribution of risk factors for childhood comorbidity in Ethiopia (DHS 2016) Diarrhea N (%) ARI N (%) Yes No Sig. Yes No Sig.
Sex of child Male 587 (5.9) 4476 (45.1) 0.05 811(8.2) 4252 (42.9) 0.855
Female 503 (5.1) 4352 (43.9) 771 (7.8) 4083 (41.2)
Place of residence
Urban 186(1.9) 1689 (16.9) 0.001 275 (2.7) 1601 (16.0) 0.001
Rural 904(9.0) 7137 (71.3) 1311 (13.1) 6736 (67.3)
Vaccination
Yes 2472 (9.1%) 2118 (54.2%) 0.001 2472(12.3%) 1993(51.0%) 0.002
No 1407(3.7%) 1264 (32.3%) 1407(5.4%) 1194(30.5%)
Antenatal visit
No 261 (3.8) 2054 (29.7) 0.001 384(5.6) 1933 (28.0) 0.029
Less 5 visits 445 (6.4) 2668 (38.6) 584 (8.4) 2531(36.6)
More 5 211(3.1) 1227(17.7) 282 (4.1) 1156 (16.7)
Wealth index
Poorest 365 (3.6) 3311 (33.1) 0.001 514(5.1) 3163(31.6) 0.001
Porrer 192 (1.9) 1469 (14.7) 305 (3.0) 1359 (13.6)
Middle 172 (1.7) 1206 (12.1) 228 (2.3) 1152 (11.5)
Richer 159 (1.6) 1055 (10.5) 225 (2.2) 990(9.9)
Richest 202 (2.0) 1785 (17.8) 314(3.1) 1673(16.7)
Mother’s Education
No 664 (6.6) 5679 (56.8) 0.001 982(9.8) 5366(53.6) 0.001
Primary 315 (3.1) 2188 (21.9) 456 (4.6) 2049 (20.5)
Secondary 74(0.7) 613 (6.1) 101(1.0) 586(5.9)
Higher 37(0.4) 346 (3.5) 47(0.5) 336(3.4)
Table 3: Cross-classification of children by diarrhea and ARI diseases in Ethiopia m (%)
Had ARI Total
No Yes
Had diarrhea No 7671(77.4%) 1156(11.7%) 8827(89.0%)
Yes 664(6.7%) 426(4.3%) 1090(11.0%)
Total
8335(84.0%) 1582(16.0%) 9917(100.0%)
488.65, df = 1, P<0.001
As summarized in Table 4, the common odds ratio (4.30) which exceeds one describes a
positive relationship between the two childhood diseases. Hence, bivariate logistic regression
model was considered suitable to investigate the related risk factors associated with diarrhea
and ARI diseases. This model provides estimates with vibrant properties of concurrence
through simultaneous estimation of effects of risk factors on the odds ratio. Besides, the
advantage of the model is that it may give better precision than unadjusted estimates
obtained by considering a multinomial distribution. The effects of risk factors on the
marginal probability of the two diseases and the odds ratio are each described with regression
equations [31].The bivariate model is parametrically dependent if Y1 and Y2 share some or
all explanatory variables, and the effects of the shared explanatory variables are jointly
estimated.
Before building the models, covariates were screened as they show some significant
association with diarrhea and ARI in the chi-square and descriptive analysis.
Maximum likelihood estimator of the model parameters obtained in R using the VGAM
by [29] and Zeligchoice by [19] packages. As reported in Table 3, in the separate model, we
found the covariates such as vaccination, mother’s education, and mother’s antenatal care
statistically significant in affecting child diarrhea status whereas residence, number of
children, vaccination, mother’s education, and wealth index were found significantly
significant factors influencing childhood acute respiratory disease status. Usually, standard
models that ignore interdependence between the two or more responses result in relatively
biased estimates [15]. Thus, we employed a joint model that captures the dependence between
childhood comorbidity in assessing the potential risk factors for diarrhea and acute
respiratory infection (ARI).
Table 4: Results for risk factors of childhood diarrhea and ARI in separate and joint
models
Variable
Marginal Model Joint Model
Diarrhea ARI Diarrhea ARI
OR SE OR SE OR SE OR SE Constants 0.0786** 0.3218 0.1182** 0.2808 0.0766** 0.3216 0.1190** 0.2806
Residence1
Residence2 1.2871 0.2470 1.7303** 0.2181 1.2865 0.2466 1.7235** 0.2178
Childsize 0.9425 0.0665 0.8788** 0.0585 0.9456 0.0664 1.0084** 0.0585
Vaccin1 1.3905** 0.1171 1.3059** 0.1006 1.4097** 0.1173 1.3034** 0.1006
Vaccin2
Educ0
Educ1 1.1618 0.1197 1.0591 0.1068 1.1595 0.1196 1.0605 0.1068
Educ2 0.6111** 0.2848 0.6008** 0.2438 0.6043** 0.2852 0.6001** 0.2439
Educ3 0.3933* 0.5408 0.6082 0.3847 0.3918** 0.5403 0.6128 0.3839
Wealth1
Wealth2 1.0660 0.1424 1.2079 0.1235 1.0844 0.1419 1.2069 0.1235
Wealth3 1.1634 0.1519 1.0823 0.1381 1.1633 0.1520 1.0817 0.1381
Wealth4 .2446 0.1654 1.4342** 0.1450 1.2472 0.1654 .4323** 0.1450
Wealth5 .0362 0.2460 1.6035** 0.2075 1.0386 0.2458 .5990** 0.2073
Antenatal1 1.1037 0.1161 1.1482 0.1006 1.1089 0.1161 1.1447 0.1005
Antenatal2 1.5853** 0.1564 1.2003 0.1446 1.6112** 0.1560 1.1938 0.1447
Antenatal3 Odds ratio 4.30
* Significant at 10% level of significance, ** Significant at 5% level of significance
In the joint response model, covariates such as vaccination, Mother’s education, antenatal
care were found significant factors for childhood diarrhea. Similarly, residence, number of
children, vaccination, mother’s education and wealth index were significant factors for acute
respiratory infection (ARI) among children age under five year. Specifically, we noticed that
children who had not received vaccinations 1.4 times more likely to develop diarrhea and
ARI. Regarding parent’s education level, children from mothers with secondary and higher
education were 0.60 and 0.39 times respectively less likely to be affected by diarrhea than
mothers with no education. These findings are consistent with a study conducted in Egypt
[21]. Furthermore, children from mothers who visited less antenatal care centers during
pregnancy were 1.61 times more likely to encounter childhood diarrhea than mother’s who
had normal antenatal care visit.
On the other hand, children living in rural areas were 1.72 times more likely to be infected
by ARI disease than their counterparts. This may be associated with inadequate health
facilities and poor lifestyle in rural areas. Regarding the number of children ever born, we
found that the likelihood of ARI infection increases, as the number of children increases i.e.
a child with more children in the family are more exposed to acute respiratory disease.
Moreover, children from mothers with secondary school education were 0.60 less likely to
be infected by ARI than mothers with no education. Regarding wealth quantile, we found
that a child from a rich family was less likely to be infected by ARI i.e. children in wealthier
families were less likely to be infected by acute respiratory disease than children in less
wealthy families. Being in a higher wealth status, compared to the lowest (poorest), was
revealed to reduce the probability of occurrence of diarrhea and ARI which is in line with
that was found in Uganda [22].
More notably, we observed that standard errors of the parameter estimates in the joint
response model were smaller compared to the corresponding standard errors from the
marginal models. This indicates efficiency gains in the joint model as compared to the
marginal models. As a result, these findings validated the need for jointly modeling of two
correlated responses.
3.1. Simulation Result
Simulation can help researchers understand the entire statistical model, take full advantage
of the parameter estimates, and convey findings in a reader-friendly manner [30].
As reported in Appendix Table 1-Table 5, for each of the two categories of the dependent
variable, the means of the predicted probabilities of an event were calculated. Then, the
absolute value of the difference between those two means was taken. If a model makes good
predictions, the cases with events should have high predicted values and the cases without
events should have low predicted values [20]. Along with the tables and graphs of the
quantities of interest in Graph 1(Appendix) indicated that the built model can make good
prediction. Each row of statistics for the expected and predicted values includes the mean
estimate for Y, the standard deviation, the median, and the 95% confidence interval measures
of uncertainty around the average.
4. Conclusion
In this paper, we specifically focused on childhood comorbidity using a joint response model
that accommodates the interdependence between the two diseases, diarrhea and ARI.
Different socio-demographic as well as other biological risk factors of diarrhea and ARI were
considered. Among 9917 under-five children from DHS (2016) in Ethiopia, 4.3% childhood
comorbidity of diarrhea and ARI was noted.A positive correlation between the two infections
was observed, and was taken into consideration in this study thereby obtaining unbiased
estimates. Preceding studies of diarrhea and ARI comorbidity modeled each disease
separately ignoring the potential correlation between the two diseases. Findings from this
study contributed to appreciate joint models that deflate parameters otherwise may be
overestimated in separate models. In the joint model, residence, vaccination, mother’s
education, and antenatal care visits during pregnancy were found significant risk factors for
diarrhea whereas residence, number of children ever born, vaccination mother’s education,
wealth index were statistically significant risk factors childhood for ARI. The two correlated
binary response variables, diarrhea, and ARI were together affected significantly by the risk
factors such as residence, vaccination, and mother’s education. As a limitation, data set from
Ethiopian DHS 2016 included most of the important covariates but spatial information was
absent, thereby we couldn’t investigate the effect of spatial variations. Future research should
look into spatial effects in order to explain the variation as a result of geographical difference
for diarrhea and ARI childhood morbidities.
Acknowledgements
The authors would like to acknowledge the Demographic and Health Survey (DHS) database
team for providing data for this study. The comments of the reviewers and the editor will be
gratefully acknowledged.
Authors’ contributions
TAB conceptualized and designed this analysis. TAB and HKG interpreted the results,
drafted and revised the manuscript. Both authors read and approved the final manuscript.
Funding
Not applicable.
Disclosure statement
No potential conflict of interest was reported by the authors.
Ethics approval and consent to participate
Not applicable.
Author details
1Department of Statistics, Haramaya University, Dire Dawa, Ethiopia
Reference
[1] C.McCulloch, Joint modelling of mixed outcome types using latent variables. Statistical
Methods in Medical Research. 2008; 17:53–73.
[2] S.L.Zeger andK.Y. Liang, Feedback models for discrete and continuous time series.
Statistica Sinica. 1991; 1:51–64.
[3] Central Statistical Agency [Ethiopia] and ICF International. Ethiopia Demographic and
Health Survey 2011; Addis Ababa, Ethiopia and Calverton, Maryland, USA: 2012;
Central Statistical Agency and ICF International.
[4] Miller, P.Nathan,A. Amouzou, M. Tafesse, E. Hazel, H. Legesse, T. Degefie, C.G.
Victora, R. E. Black, and J. Bryce, Integrated Community Case Management of
Childhood Illness in Ethiopia: Implementation Strength and Quality of Care. The
American Journal of Tropical Medicine and Hygiene.2014; 13:751.
[5] World Health Organization (WHO) and UNICEF. Ending Preventable Child Deaths from
Pneumonia and Diarrhea by 2025: The Integrated Global Action Plan for Pneumonia and
Diarrhea (GAPPD). Geneva, Switzerland:2013; WHO and UNICEF.
[6] N.B.Kandala,G.Gebrenegus, Ageo-addictive bayesian discretetime survival model and its
application to spatial analysis of childhood mortality in malawi. Quality and Quantity
2006; 40: 935-957.
[7] N.B.Kandala, M.A.Magadi, andN.J.Madise, An investigation of district spatial variations
of childhood diarrhoea and fever morbidity in Malawi. Soc Sci Med 2006; 62: 1138-
1152.
[8] K.Khatab and N.B. Kandala, Latent variable modelling of risk factors associated with
childhooddiseases: Case study for Nigeria', Asian Pacific Journal of Tropical Disease.
Asian Pacific Journal of Tropical Disease.2011; 2222–1808:169–176.
[9] L.N.Kazembe, A.S.Muula, C.C.Appleton, I. Kleinschmidt, Modelling the effect of
malaria endemicity on spatial variations in childhood fever, diarrhea and pneumonia in
Malawi. Int. J. Health Geogr. 2007, 6, 33–43.
[10] C.G.Victora, T.Adam, J.Bryce, andD.B. Evans, Integrated Management of the Sick;
Child Disease Control Priorities Project: Washington, DC, USA, 2006. Available Online:
http://www.dcp2.org/pubs/DCP/63/Section/9372 (accessed on 15 July 2019).
[11] B.Fenn, S. Morris, andR.E. Black, Comorbidity inchildhood in Ghana: Magnitude,
associated factors and impact on mortality. Int. J. Epidemiol. 2005, 34, 368–375.
[12] L.N.Kazembe and J.J. Namangale, A Beyesian multinomial model to analyse spatial
patterns of childhood co-morbidity in Malawi. Eur. J. Epidemiol. 2007, 22, 545–556.
[13] TW. Yee,Vector Generalized Linear and Additive Models: With an Implementation in
R. 2015; Springer, New York, USA.
[14] E. Bbaale, Determinants of diarrhea and acute respiratory infection among under-fives
in Uganda. Australas Med J. 2011; 4(7): 400–409.
[15] M. Ghebremikael, Joint modeling of correlated binary out comes: HIV-1and HSV-2 co-
infection. Journal of Applied Statistics 2015, 42:2180
.https://doi.org/10.1080/02664763.2015.1022138
[16] J. R. Dale, Global cross-ratio models for bivariate, discrete, ordered responses.
Biometrics1986;42: 909-917.
[17] Skrondal and S. Rabe-Hesketh,Generalized Latent Variable Modeling, Chapman & Hall,
New York, 2004.
[18] T. W. Yee and T. J. Hastie, Reduced-rank vector generalized linear models.
StatisticalModelling, 2003; 3:15-41.
[19] Kosuke Imai, Gary King, and Olivia Lau.. \Zelig: Everyone's Statistical. Software2007;
http://GKing.harvard.edu/zelig.
[20] T. Tjur, Coefficients of determination in logisticregression models—A new proposal:
The coefficient of discrimination. The American Statistician63: 2009; 366-372.
[21] AH. El-Gilany and S. Hammad, Epidemiology of diarrheal diseases among children
under age 5 years in Dakahlia, Egypt. Eastern Mediterranean Health Journal.
2000;11(4): 762-775.
[22] E. Bbaale, Determinants of diarrhea and acute respiratory infection among under-fives
in Uganda. AMJ 2011, 4, 7, 400-409 http//dx.doi.org/10.4066/AMJ.2011.723
[23] P.McCullagh and J A.Nelder,Generalized Linear Models (second edition). London,
United Kingdom: Chapman & Hall; 1989.
[24] R.E. Black,S.S.Morris, andJ. Bryce, Where and why are 10 million children dying every
year?Lancet, 2003; 361(9376), 2226–2234, DOI 10.1016/s0140-6736(03)13779-8.
[25] M.Kosek, C. Bern, andR.L. Guerrant, The global burden of diarrhoeal disease, as
estimated from studiespublished between 1992 and 2000.Bull. World Health Organ,
2003; 81(3), 197–204.
[26] A.Downing, D.Forman,M.S.Gilthorpe, K.L.Edwards, andS.O.Manda, Joint disease
mapping using six cancers in the Yorkshire region of England. Int. J. Health
Geogr. 2008;7:41–55. doi: 10.1186/1476-072X-7-41.
[27] J. Palmgren, Regression Models for Bivariate Binary Responses. Technical Report no.
101, Department of Biostatistics, University of Washington, Seattle.1989.
[28] S. Cessie, andJ. C. Houwelingen, Logistic regression for correlated binary data. Applied
Statistics, 1994; 43, 95–108.
[29] T. WYee, and T. Dirnbock, Models for analysing species’ presence/absence data at two
time points. Journal of Theoretical Biology 2009; 259, 684-694.
[30] Fair, Ray C, Estimating the Expected Predictive Accuracy of Econometric Models.
International Economic Review 1980; 21:355–378.
[31] Abud Darda Ghazanfar Ali. Modelling of African Farm Dynamics Using Bivariate
Binary Logistic Regression in WinBUGS. Mater Thesis. 2009; Lund Univesity.
[32] McCullagh, P. and Nelder, J. A. Generalized Linear Models. 2nd ed. 1989; London:
Chapman & Hall.
Appendices
1. Quantities of interest from bivariate logit model Simulation for x(lowest value of explanatory variable)
Table 1: Expected values
mean sd 50% 2.5% 97.5%
Pr(Y1=0, Y2=0) 0.74854988 0.014409098 0.74913524 0.71984276 0.77557475 Pr(Y1=0, Y2=1) 0.12278475 0.010412208 0.12256883 0.10387570 0.14359428 Pr(Y1=1, Y2=0) 0.07540105 0.007805833 0.07522129 0.06004650 0.09160451 Pr(Y1=1, Y2=1) 0.05326432 0.006478022 0.05286303 0.04232368 0.06682350
Table 2: Predicted values
0 1
(Y1=0, Y2=0) 0.289 0.711 (Y1=0, Y2=1) 0.843 0.157 (Y1=1, Y2=0) 0.888 0.112 (Y1=1, Y2=1) 0.980 0.020
Simulation for x1(highest value of the explanatory variable) Table 3: Expected values
mean sd 50% 2.5% 97.5%
Pr(Y1=0, Y2=0) 0.83784969 0.04727187 0.84443911 0.729413394 0.9124655 Pr(Y1=0, Y2=1) 0.10031222 0.03759618 0.09337025 0.046248931 0.1913946 Pr(Y1=1, Y2=0) 0.04132368 0.02283666 0.03550673 0.011918510 0.1019103 Pr(Y1=1, Y2=1) 0.02051441 0.01307836 0.01775972 0.005519475 0.0505012
Table 4: Predicted values
0 1
(Y1=0, Y2=0) 0.160 0.840 (Y1=0, Y2=1) 0.897 0.103 (Y1=1, Y2=0) 0.951 0.049 (Y1=1, Y2=1) 0.992 0.008
Table 5: Fist difference
mean sd 50% 2.5% 97.5% Pr(Y1=0,Y2=0) 0.08929981 0.04562680 0.09516639 -0.01514334 0.16144689 Pr(Y1=0,Y2=1) -0.02247253 0.03606915-0.02729431 -0.07446998 0.06537456 Pr(Y1=1,Y2=0) -0.03407737 0.02257006 0.03796583 -0.06516763 0.02111796 Pr(Y1=1,Y2=1) -0.03274992 0.01293345-0.03455233 -0.05159941-0.00274422
(a) (b)
(c)
Fig 1: Graphs of quantities of interest for Diarrhea model (a), ARI model (b) and Joint
model(c)